Domain decomposition techniques have been successfully used in the context of ML applied to PDEs [1]}. These techniques enable local learning from smaller restricted domains and have proved to accelerate learning of neural networks. In [2]}, [1]}, the authors proposed a hybrid method combining ML with domain decomposition to reduce the computational cost of finite element solvers. Similarly, domain decomposition has been extensively used in the context of physics informed neural networks to reduce training costs by enabling distributed training on multiple GPUs [4]}, [5]}. [6]} and [7]} learn on localized domains but infer on larger computational domains using a stitching process. [8]} and [9]} learn coefficients of numerical discretization schemes from high fidelity data, which is sub-sampled on coarse grids. [10]} learn surrogate models for smaller components to allow for cheaper simulations. [11]} learn local prolongation operators from discretization matrices, to improve the rate of convergence of the multigrid linear solvers. Alternatively, other methods such as [12]}, [13]}, [14]} make us of graph-based models to improve generalization by learning on local elements. In this work, we use a Cartesian grid to discretize each subdomain. However, GNNs [15]}, [16]}, [17]}, [18]}, [19]} as well as FEM use other discretizations such as triangle or polygonal meshes. In future, these ideas can be used to handle unstructured meshes in subdomains. Additionally, other ML methods improve the learning of ML models by compressing PDE solutions on to lower-dimensional manifolds. This has shown to improve accuracy and generalization capability of neural networks [20]}, [21]}, [22]}, [23]}, [24]}, [25]}. Although, here we propose an iterative inferencing approach in the context of solving PDEs, it is inspired from the implicit deep learning approach used in a variety of machine learning tasks [26]}, [27]}, [28]}, [29]}. In this work, domain decomposition is combined with latent space learning is employed to accurately represent solutions on local subdomains and allow scalability to bigger computational domains.
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